Chapter 8
Turbulence
SUMMARY: Almost all environmental fluid flows, be they of air or water, are turbulent. Unfortunately, the highly intermittent and irregular character of turbulence
defies analysis, and there does not yet exist a unifying theory of turbulence, not
even one for its statistical properties. So, the approach in this chapter is necessarily
much more empirical and heuristic. Each section considers turbulence in on of its
most common manifestations in natural fluid flows: homogeneous turbulence, shear
turbulence, and turbulence in stratified fluids. It then explores several classical
methods to model these types of turbulence.
8.1
Homogeneous and Isotropic Turbulence
At a very basic level, a turbulence flow can be interpreted as a population of many
eddies (vortices), of different sizes and strengths, embedded in on another and forever changing, giving a random appearance to the flow (Figure 8.1). Two variables
then play a fundamental role: d, the characteristic diameter of the eddies, and ů,
their characteristic orbital velocity. Since the turbulent flow consist in many eddies,
of varying sizes and speeds, ů and d do not assume each of a single value but vary
within a certain range. In stationary, homogeneous and isotropic turbulence, that
is, a turbulent flow that statistically appears unchanging in time, uniform in space
and without preferential direction, all eddies of a given size (same d) behave more or
less in the same way and can be thought of sharing the same characteristic velocity
ů. In other words, we make the assumption that ů is a function of d (Figure 8.2).
8.1.1
Energy cascade
In the view of Kolmogorov (1941), turbulent motions span a wide range of scales
ranging from a macroscale at which the energy is supplied, to a microscale at which
energy is dissipated by viscosity. The interaction among the eddies of various scales
135
136
CHAPTER 8. TURBULENCE
Figure 8.1: Drawing of a turbulent flow by Leonardo da
Vinci (1452–1519), who recognized that turbulence involves
a multitude of eddies at various
scales.
passes energy sequentially from the larger eddies gradually to the smaller ones. This
process is known as the turbulent energy cascade (Figure 8.3).
If the state of turbulence is statistically steady (statistically unchanging turbulence intensity), then the rate of energy transfer from one scale to the next must
be the same for all scales, so that no group of eddies sharing the same scale sees its
total energy level increase or decrease over time. It follows that the rate at which
energy is supplied at the largest possible scale (dmax ) is equal to that dissipated at
the shortest scale (dmin ). Let us denote by ǫ this rate of energy supply/dissipation,
per unit mass of fluid:
ǫ
= energy supplied to fluid per unit mass and time
= energy cascading from scale to scale, per unit mass and time
= energy dissipated by viscosity, per unit mass and time.
The dimensions of ǫ are:
M L2 T −2
= L2 T −3 .
(8.1)
MT
With Kolmogorov, we further assume that the characteristics of the turbulent
eddies of scale d depend solely on d itself and on the energy cascade rate ǫ. This
is to mean that the eddies know how big they are, at which rate energy is supplied
to them and at which rate they must supply it to the next smaller eddies in the
cascade. Mathematically, ů depends only on d and ǫ. Since [ů] = LT −1, [d] = L
and [ǫ] = L2 T −3 , the only dimensionally acceptable possibility is:
[ǫ] =
ů(d) = A(ǫd)1/3 ,
in which A is a dimensionless constant on the order of unity.
(8.2)
8.1. HOMOGENEOUS AND ISOTROPIC TURBULENCE
137
Figure 8.2: Eddy orbital velocity versus eddy length scale in homogeneous turbulence. The largest eddies spin the fastest.
Thus, the larger ǫ, the larger ů. This makes sense, for a greater energy supply to
the system generates stronger eddies. Equation (8.2) further tells that the smaller
d, the weaker ů. This could not have been anticipated and must be accepted as
a result of the theory. The implication is that the smallest eddies have the lowest
speeds, while the largest ones have the highest speeds and thus contain the bulk of
the kinetic energy.
8.1.2
Largest and shortest length scales
Typically, the largest possible eddies in the turbulent flow are those that extend
across the entire system, from boundary to opposite boundary, and therefore
dmax = L,
(8.3)
where L is the geometrical dimension of the system (such as the width of the domain
or the cubic root of its volume). In environmental flows, there is typically a noticeable scale disparity between a relatively short vertical extent (depth, height) and
a comparatively long horizontal extent (distance, length) of the system. Examples
are:
rivers :
atmosphere :
depth << width << length
height << physically relevant horizontal distances.
In such situations, we must clearly distinguish eddies that rotate in the vertical
plane (about a horizontal axis) from those that rotate horizontally (about a vertical
axis). In rivers, we may furthermore distinguish transverse eddies from longitudinal
eddies.
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CHAPTER 8. TURBULENCE
Figure 8.3: The turbulent energy cascade. According to this theory, the energy
fed by external forces excites the largest possible eddies and is gradually passed to
ever smaller eddies, all the way to a minimum scale where this energy is ultimately
dissipated by viscosity.
The shortest eddy scale is set by viscosity, because the shorter the eddy scale, the
stronger the velocity shear and the more important the effect of viscosity. Consequently, the shortest eddy scale can be defined as the length scale at which viscosity
becomes dominant. Viscosity, denoted by ν, has for dimensions1 :
[ν] = L2 T −1 .
If we assume that dmin depends only on ǫ, the rate at which energy is supplied to that
scale, and on ν, because those eddies sense viscosity, then the only dimensionally
acceptable relation is:
dmin ∼ ν 3/4 ǫ−1/4 .
(8.4)
Therefore, dmin depends on the energy level of the turbulence. The stronger the
turbulence (the bigger ǫ), the shorter the minimum length scale at which it is capable
of stirring. The quantity ν 3/4 ǫ−1/4 , called the Kolmogorov scale, is typically on the
order of a few millimeters or shorter.
The span of length scales in a turbulent flow is related to its Reynolds number.
Indeed, in terms of the largest velocity scale, which is the orbital velocity of the
largest eddies, U = ů(dmax ) = A(ǫL)1/3 , the energy supply/dissipation rate is
ǫ =
1 Values
U3
U3
∼
,
3
A L
L
(8.5)
for ambient air and water are: νair = 1.51 × 10−5 m2 /s and νwater = 1.01 × 10−6 m2 /s.
8.1. HOMOGENEOUS AND ISOTROPIC TURBULENCE
139
Figure 8.4: A real water cascade, showing the tumbling
down of water from the highest to the lowest point. In
analogy, energy of a turbulent
flow is tumbling down from the
largest to the shortest eddy
c
scale. [Photo Ian
Adams]
and the length scale ratio can be expressed as
L
dmin
L
ν 3/4 ǫ−1/4
LU 3/4
∼
ν 3/4 L1/4
∼ Re3/4 ,
∼
(8.6)
where Re = U L/ν is the Reynolds number of the flow. As we could have expected,
a flow with a higher Reynolds number contains a broader range of eddies.
Example 8.1 Atmospheric turbulence
Consider the atmospheric boundary layer, spanning a height of about 1000 m
above the ground. If the typical wind speed is 10 m/s, then the Reynolds number
can be estimated to be
Re =
(10 m/s)(1000 m)
UL
=
= 6.6 × 108 .
ν
(1.51 × 10−5 m2 /s)
which yields Re3/4 = 4.1 × 106 and
dmin ∼
L
= 2.4 × 10−4 m
Re3/4
or about 0.24 mm.
The energy supply/dissipation rate is estimated to be
ǫ ∼
U3
(10 m/s)3
=
= 1 m2 /s3 .
L
(1000 m)
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CHAPTER 8. TURBULENCE
or 1 Joule per kilogram of air and per second.
8.1.3
Energy spectrum
In turbulence theory, it is customary to consider the so-called power spectrum,
which is the distribution of kinetic energy per mass across the various length scales.
For this, we need to define a wavenumber. Because velocity reverses across the
diameter of an eddy, the eddy diameter should properly be considered as half of the
wavelength:
k =
2π
π
=
.
wavelength
d
(8.7)
The extremal values are kmin = π/L and kmax ∼ ǫ1/4 ν −3/4 .
The kinetic energy E per mass of fluid has dimensions of ML2 T−2 /M = L2 T−2 .
The fraction dE contained in the eddies with wavenumbers ranging from k to k + dk
is defined as
dE = Ek (k) dk.
It follows that the dimension of Ek is L3 T−2 , and dimensional analysis prescribes:
Ek (k) = B ǫ2/3 k −5/3 ,
(8.8)
where B is a second dimensionless constant. It can be related to A of Equation
(8.2) because the integration of Ek (k) from kmin = π/L to kmax ∼ ∞ is the total
energy in the system, which in good approximation is that contained in the largest
eddies, namely U 2 /2. Thus,
Z ∞
U2
,
(8.9)
Ek (k) dk =
2
kmin
from which follows
2
1 2
A .
(8.10)
B =
2/3
2
3π
The value of B has been determined experimentally and found to be about 1.5
(Pope, 2000, page 231). From this, we estimate A to be 0.97.
The −5/3 power law of the energy spectrum has been observed to hold well in
the inertial range, that is, for those intermediate eddy diameters that are remote
from both largest and shortest scales. Figure 8.5 shows the superposition of a large
number of longitudinal power spectra2 . The straight line where most data overlap
in the range 10−4 < kν 3/4 /ǫ1/4 < 10−1 corresponds to the −5/3 decay law predicted
by the Kolmogorov turbulent cascade theory. The higher the Reynolds number of
the flow, the broader the span of wavenumbers over which the −5/3 law holds.
2 The longitudinal power spectrum is the spectrum of the kinetic energy associated with the
velocity component in the direction of the wavenumber.
8.2. SHEAR-FLOW TURBULENCE
141
The few crosses at the top of the plot, which extend a set of crosses buried in the
accumulation of data below, correspond to data in a tidal channel (Grant et al.,
1962), for which the Reynolds number was the highest.
Figure 8.5: Longitudinal power spectrum of turbulence calculated from numerous
observations taken outdoor and in the laboratory. [From Saddoughi and Veeravalli,
1994]
There is, however, some controversy over the −5/3 power law for Ek . Some investigators (Saffman, 1968; Long, 1997 and 2003) have proposed alternative theories
that predict a −2 power law.
8.2
Shear-Flow Turbulence
Most environmental fluid systems are much shallower than they are wide. Such
are the atmosphere, oceans, lakes and rivers. Their vertical confinement forces the
flow to be primarily horizontal, and the vertical velocity, if any, is relatively weak.
The ratio of vertical to horizontal velocity is typically on the order of the geometric
142
CHAPTER 8. TURBULENCE
Figure 8.6: Velocity distribution across a sample of natural and artificial channels.
[From White, 2003]
aspect ratio of the domain (depth over width), or less.
Unavoidable in such shallow–wide situation is friction between the main horizontal motion and the bottom boundary. Friction acts to reduce the velocity from
some finite value in the interior of the flow to zero at the bottom, thus creating a
vertical shear. Mathematically, if u is the velocity component in one of the horizontal directions and z the elevation above the bottom, then u is a function of z.
The function u(z) is called the velocity profile and its derivative du/dz, the velocity
shear. Figure 8.6 sketches a few examples, which show velocity distributions across
several water channels.
Environmental flows are invariably turbulent (high Reynolds number) and this
greatly complicates the search for the velocity profile. As a consequence, much of
what we know is derived from observations of actual flows, either in the laboratory
or in nature.
The turbulent nature of the shear flow along a smooth or rough surface includes
variability at short time and length scales, and the best observational techniques
for the detailed measurements of these have been developed for the laboratory
rather than outdoor situations. Figures 8.7 and 8.8 show the details of a turbulent
flow along a smooth straight wall. Note the rolling over of fluid particles across
the primary direction of the flow. Such cross-flow exchanges are responsible for
momentum transfer in the direction perpendicular to the boundary, which sets the
average velocity profile.
Numerous laboratory measurements of turbulent flows along smooth straight
surfaces have led to the conclusion that the velocity varies solely with the stress τb
exerted against the bottom, the fluid molecular viscosity µ, the fluid density ρ and,
143
8.2. SHEAR-FLOW TURBULENCE
Figure 8.7: Near-wall structure
of a turbulent shear flow visualized by hydrogen bubbles and
viewed from the side. The pair
of photographs shows the formation of a streamwise vortex
in a matter of a few seconds.
(From Kline et al., 1967)
of course, the distance z above the bottom. Thus,
u(z) = F (τb , µ, ρ, z).
Dimensional analysis permits the elimination of the mass dimension shared by τb ,
µ and ρ but not present in u and z, and we may write more simply:
u(z) = F
τb µ
, ,z .
ρ ρ
The ratio µ/ρ is the kinematic viscosity ν (units of m2 /s), whereas the ratio
τb /ρ has the same dimension as the square of a velocity (units of m2 /s2 ). For
convenience, it is customary to define
u∗ =
r
τb
,
ρ
(8.11)
which is called the friction velocity or turbulent velocity. Physically, its value is
related to the orbital velocity of the vortices that create the cross-flow exchange of
particles and the momentum transfer.
The velocity structure thus obeys a relation of the form
u(z) = F (u∗ , ν, z),
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CHAPTER 8. TURBULENCE
Figure 8.8: Top view of the same laboratory experiment as shown in Figure 8.7. The
flow is directed from top to bottom in the photograph, and hydrogen bubbles are
generated along the horizontal line. Hydrogen bubbles form streaks, which indicate
a pattern of alternating convergence and divergence in the cross-stream direction.
This pattern is the horizontal manifestation of streamwise vortives. (From Kline et
al., 1967)
and further use of dimensional analysis allows us to reduce this to a function of a
single variable:
u z u(z)
∗
.
= f
u∗
ν
Logarithmic profile
The observational determination of the function f has been repeated countless
times, every time with the same results, and it suffices here to provide a single
report (Figure 8.9). When the velocity ratio u/u∗ is plotted versus the logarithm
of the dimensionless distance u∗ z/ν, not only do all the points coalesce on a single
curve, confirming that there is indeed no other variable to be invoked, but the curve
also behaves as a straight line over a range of two orders of magnitude (from u∗ z/ν
between 101 and 103 ).
If the velocity is linearly dependent on the logarithm of the distance, then we
can write for this portion of the velocity profile:
u(z)
u∗ z
= A ln
+ B.
u∗
ν
Numerous experimental determinations of the constants A and B provide A = 2.44
and B = 5.2 within a 5% error (Pope, 2000). Tradition has it to write the function
as:
u(z) =
u∗ z
u∗
ln
+ 5.2 u∗ ,
κ
ν
(8.12)
145
8.2. SHEAR-FLOW TURBULENCE
Figure 8.9: Mean velocity profiles in fully developed turbulent
channel flow measured by Wei
and Willmarth (1989) at various Reynolds numbers: circles
Re = 2970, squares Re = 14914,
upright triangles Re = 22776,
and downright triangles Re =
39582. The straight line on this
log-linear plot corresponds to
the logarithmic profile of Equation (8.12). (From Pope, 2000)
where κ = 1/A = 0.41 is called the von Kármán constant3
The portion of the curve closer to the wall, where the logarithmic law fails,
may be approximated by the laminar solution. Constant laminar stress νdu/dz =
τb /ρ = u2∗ implies u(z) = u2∗ z/ν there. Ignoring the region of transition in which the
velocity profile gradually changes from one solution to the other, we can attempt to
connect the two. Doing so yields u∗ z/ν = 11. This sets the thickness of the laminar
boundary layer δ as the value of z for which u∗ z/ν = 11, i.e.
δ = 11
ν
.
u∗
(8.13)
Most textbooks (e.g. Kundu, 1990) give δ = 5ν/u∗ , for the region in which the
velocity profile is strictly laminar, and label the region between 5ν/u∗ and 30ν/u∗
as the buffer layer, the transition zone between laminar and fully turbulent flow.
For water in ambient conditions, the kinetic molecular viscosity ν is equal to 1.0
× 10−6 m2 /s, while the friction velocity in a typical river rarely falls below 1 cm/s.
This implies that δ can hardly exceed 1 mm in a river and is almost always smaller
than the height of the cobbles, ripples and other asperities that typically line the
bottom of the channel.
When this is the case, the velocity profile above the bottom asperities no longer
depends on the molecular viscosity of the fluid but on the so-called roughness height
zo , such that
3 in honor of Theodore von Kármán (1881–1963), Hungarian-born physicist and engineer who
made significant contributions to fluid mechanics while working in Germany and who first introduced this notation.
146
CHAPTER 8. TURBULENCE
Figure 8.10: Velocity profile in
the vicinity of a rough wall. The
roughness heigh zo is smaller
than the averaged height of the
surface asperities. So, the velocity u falls to zero somewhere
within the asperities, where local flow degenerates into small
vortices between the peaks, and
the negative values predicted by
the logarithmic profile are not
physically realized.
u(z) =
u∗
z
,
ln
κ
zo
(8.14)
as depicted in Figure 8.10. It is important to note that the roughness height is
not the average height of bumps on the surface but is equal to a small fraction
of it, about one tenth (Garratt, 1992, page 87). Table 8.1 lists a few values of
environmental relevance. In this list, d90 is the particle diameter such that 90% of
the particles have a smaller diameter than this.
Table 8.1 Typical values of the roughness height
147
8.2. SHEAR-FLOW TURBULENCE
Type of surface
zo
(m)
Reference
Sand, snow or sea
Active sand sheet
Sand sheet
Gravel lag
Bare soil
Flat open country
Grass - sparse
Grass - thick
Grass - thin
Wheat field
Corn field
Vines - along rows
Vines - across rows
Vegetation
Savannah
Scrub
Trees
Trees - sparse
Forest - temperate
0.016u2∗ /g
0.00004
0.0004–0.0005
0.0002
0.001–0.01
0.02–0.06
0.0012
0.026
0.05
0.015
0.064
0.023
0.12
0.2
0.4
0.049
0.4
0.1 of tree height
0.28–0.92
Forest - tropical
2.2–4.8
Residential neighborhood
City with high rises
Hilly terrain
1.0–5.0
Chamberlain (1983)
Draxler et al. (2001)
Draxler et al. (2001)
Draxler et al. (2001)
Garratt (1992)
Davenport (1965)
Clarke et al. (1971)
Su et al. (2001)
Sutton (1953)
Garratt (1977)
Kung (1961)
Hicks (1973)
Hicks (1973)
Fichtl and McVehil (1970)
Garratt (1980)
Su et al. (2001)
Fichtl and McVehil (1970)
Guyot and Seguin (1978)
Hicks et al. (1975),
Thom et al. (1975),
Jarvis et al. (1976)
Thomson and Pinker (1975),
Shuttleworth (1989)
Davenport (1965)
Concrete channel
Mud
Coarse grains
Rocks
0.0003–0.003
0.0
2.0 d90
0.05
Atmosphere
1–10
Water
Chanson (2004)
Kamphuis (1974)
Drag coefficient
The average velocity in a vertically confined domain can be related to the bottom
stress. For a rough bottom at z = 0 and a free surface at z = h, the average velocity
is given by:
ū =
=
Z h
Z h
u∗
z
1
dz
u(z)dz =
ln
h 0
κh 0
zo
h
u∗
ln
− 1 ,
κ
zo
(8.15)
which permits to relate the friction velocity u∗ to the average velocity ū:
u∗ =
κū
.
ln(h/zo ) − 1
(8.16)
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CHAPTER 8. TURBULENCE
This in turn provides the relationship between the bottom stress and the average
velocity:
τb = ρu2∗ =
ρκ2 ū2
.
[ln(h/zo ) − 1]2
(8.17)
If we introduce a drag coefficient CD such that τb = CD ρū2 , then its value is:
−2
h
CD = κ2 ln
−1
.
zo
(8.18)
Note that the drag coefficient is not a constant but depends on the ratio of the fluid
depth to the roughness height. For κ = 0.41 and a ratio h/zo in the range 50–1000,
the drag coefficient varies between 0.005 and 0.020.
8.3
Mixing Length
To solve more general problems in turbulence, an attempt has been made to assimilate the mixing caused by turbulence to an enhanced viscosity. This amounts to a
search for a turbulent viscosity (often called an eddy viscosity) that would replace in
turbulent flows the molecular viscosity of laminar flows. In analogy with Newton’s
law for viscous fluids, which has the tangential stress τ proportional to the velocity
shear du/dz with the coefficient of proportionality being the molecular viscosity µ,
one writes for turbulent flow:
du
,
(8.19)
dz
where the turbulent viscosity µT supersedes the molecular viscosity µ.
For the logarithmic profile (8.14) of a flow along a rough surface, the velocity
shear is du/dz = u∗ /κz and the stress τ is uniform across the flow (for lack of
acceleration and of other forces): τ = τb = ρu2∗ , giving
τ = µT
ρu2∗ = µT
u∗
κz
and thus
µT = ρκzu∗ .
(8.20)
Note that unlike the molecular viscosity, the turbulent viscosity is not constant in
space, for it is not a property of the fluid but of the flow, including its geometry.
The corresponding turbulent kinematic viscosity is
νT =
µT
= κzu∗ ,
ρ
which can be expressed as the product of a length by the turbulent velocity:
149
8.3. MIXING LENGTH
νT = l m u ∗ ,
(8.21)
with the mixing length lm defined as
lm = κz,
(8.22)
for the turbulent flow along a rough boundary.
To generalize to turbulent flows other than the logarithmic profile, we keep
Equation (8.21) but intend to adapt the mixing length lm and turbulent velocity u∗
to every situation. The turbulent velocity is replaced by means of the velocity shear
by reasoning that it is the shear that creates flow instabilities, induces turbulence
and thus creates a turbulent viscosity. In other words, the greater the velocity
shear is, the larger the turbulent viscosity ought to be. Let us then eliminate the
turbulent velocity u∗ in favor of the velocity shear du/dz and then generalize the
latter for flows other than parallel flow, as follows:
du
du
= ρνT
dz
dz
with the local turbulent velocity defined from the magnitude of the local stress by
|τ | = ρu2∗ and νT = lm u∗ according to (8.21):
du du 2
ρu∗ = ρlm u∗ −→
u∗ = lm dz
dz
τ = µT
yielding
2 du νT = l m u ∗ = l m
dz .
(8.23)
In arbitrary, three-dimensional turbulent flows there are several components
to the velocity shear, actually an entire array. Smagorinsky (1963) proposed the
following extension. First, the rate-of-strain tensor is defined, with components
∂uj
1 ∂ui
(8.24)
+
Sij =
2 ∂xj
∂xi
where i and j are indices running from 1 to 3, such that (x1 , x2 , x3 ) = (x, y, z)
and (u1 , u2 , u3 ) = (u, v, w). From the tensor Sij is defined the overall strain S by
S2 = 2
3
3
X
X
i=1
2
Sij
(8.25)
j=1
that supersedes the earlier velocity shear du/dz. According to this model of turbulence, the turbulent kinematic viscosity is
2
νT = l m
S.
(8.26)
But, this way of proceeding includes not only the actual velocity shear components (such as ∂u/∂y, ∂v/∂x, etc.) but also convergence/divergence terms (∂u/∂x,
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CHAPTER 8. TURBULENCE
∂v/∂y and ∂w/∂z) that do not contribute to instabilities and turbulence. A remedy
to this situation is the model of Baldwin and Lomax (1978) in which the rate-ofrotation tensor
Ωij =
1
2
∂ui
∂uj
−
∂xj
∂xi
(8.27)
is used instead of the rate-of-strain tensor. All non-zero elements of this tensor
correspond to vorticity components. From the tensor Ωij is defined the overall
shear Ω by
Ω2 = 2
3
3
X
X
i=1
Ω2ij
(8.28)
j=1
and, according to this model, the turbulent kinematic viscosity is
2
Ω.
νT = l m
(8.29)
It remains now to say something about the mixing length lm . Since lm = κz
for parallel flow along a rough boundary, where z is the distance to the boundary,
one possibility is to take for the mixing length the product of the von Kármán
constant κ = 0.41 by the distance to the nearest wall. For a flow between two
parallel boundaries, like a river between its two banks (at y = 0 and y = W ), a
choice is
lm = κ
8.4
y(W − y)
.
W
(8.30)
Turbulence in Stratified Fluids
The presence of stratification induces buoyancy forces against which turbulent motions need to work. The result is a subdued form of turbulence. Turner (1973,
Chapter 5), Thorpe (review, JGR, 1987), Tritton (1988, Section 21.7).
Ozmidov buoyancy vertical scale
Lb =
r
ǫ
N3
(8.31)
in which ǫ is the energy dissipation rate per unit mass and N is the stratification
frequency.
8.5. TWO-DIMENSIONAL TURBULENCE
8.5
151
Two-dimensional Turbulence
Because of their small vertical-to-horizontal aspect ratio, environmental systems
exhibit in the horizontal a form of turbulence that is nearly two-dimensional.
8.6
Closure Schemes
Since no complete theory of turbulence exists, there is a need to distill somehow
the results of observations into some empirical rules. A computer simulation model
that incorporates one or several of these rules is said to include a closure scheme.
A large number of closure schemes have been proposed over the years, with varying
degrees of success. We present here only a couple of them, which have each been
tested extensively in the context of environmental systems.
k − ǫ model.
Mellor and Herring (1973), Mellor and Yamada (1982).
8.7
Large-Eddy Simulations
Pope (2000, Chapter 13)
Problems
8-1. What would be the energy spectrum Ek (k) in a turbulent flow where all length
scales were contributing equally to dissipation? Is this spectrum realistic?
8-2. The earth receives 1.75 × 1017 W from the sun, and we can assume that half
of this energy input is being dissipated in the atmosphere, with the rest going
to land and sea.
(a) Using the known ground-level pressure and surface area of the earth, determine the mass of the atmosphere.
(b) Estimate the rate of energy dissipation ǫ in the atmosphere.
(c) What is the Reynolds number of the atmosphere?
(c) Finally, estimate the smallest eddy scale in the air and its ratio to the
largest scale.
Useful numbers: Standard atmospheric pressure is 101,325 Pa, the radius of
the earth is 6,371 km, and the molecular viscosity of air at ambient temperature and pressure is ν = 1.51 × 10−5 m2 /s.
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CHAPTER 8. TURBULENCE
8-3. There are applications in turbulent boundary-layer flow for which the logarithmic function creates mathematical complications and it is desirable to use
a power law to represent the velocity profile, such as
u(z) = U
= U
z α
for z < d
d
for d ≤ z,
with d being a suitable boundary-layer thickness. Determine the values of d
and the exponent α for best fit over the interval 10zo < z < 2000zo. What
is then the relation between the far-field velocity U and the friction velocity
u∗ ? And, from this, establish an approximate drag law relating wall stress to
far field velocity.
8-4. A 20-m deep reservoir has a river throughflow sustaining a steady current
varying from 12 cm/s near the surface to zero at the bottom. At the time
of these observations, a wind was blowing that imposed a stress of 0.1 N/m2
on the water surface. Compare the turbulence activity generated by the wind
stress to that maintained by the river flow. Which one by itself leads to the
largest dissipation rate? What is the Kolmogorov dissipation scale in the
reservoir?
8-5.
8-6.
8-7.